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Mathematics for the Digital Age

Programming in Python

Maria Litvin Phillips Academy, Andover, Massachusetts

Gary Litvin

Skylight Software, Inc.

Skylight Publishing Andover, Massachusetts

and

Skylight Publishing 9 Bartlet Street, Suite 70 Andover, MA 01810 web: http://www.skylit.com e-mail: sales
skylit.com

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Printed in the United States of America

331

Index (infinity symbol), 64 – (difference) operator for sets, 252 != operator, 69 “”” literal string delimiter, 32, 33 & (intersection) operator for sets, 252 & operator, 123, 125 *= operator, 68 / operator, 53 // operator, 53 ^ (symmetric difference) operator for

sets, 253 ^ operator, 123 | (union) operator for sets, 252 | operator, 123 ~ operator, 123 + operator for lists, 154 += operator, 68 , = operators, 69 > operators, 126 -= operator, 68 == operator, 69 absolute pathname, 161 add method of set, 253 adder circuit, 122 addition rule for probabilities, 233 adjacency matrix, 280 adjacent vertices in a graph, 279 algorithm, 11 anagram, 137, 166 analog electronics, 117 AND circuit, 116 AND gate, 118 and operator, 69, 107

arguments, 2, 51 arithmetic mean, 54, 59 arithmetic sequence, 58 common difference, 59 ASCII code, 23 ASCII file, 160 Assembly Language, 20 assignment operator, 38 augmented assignment operators, 68 backslash, 33 base case in recursion, 196 base-10 numbers, 78 bin built-in function, 84 binary arithmetic, 84 binary files, 160 binary system, 81 binomial coefficients, 215 binomial theorem, 215 bitwise logical operators, 123 Blackjack, 231 Boole, George, 95 Boolean algebra, 95 Boolean functions, 109 bucket (in a hash table), 251 built-in functions, 39 bytecode, 25 carriage return, 160 Central Processing Unit (CPU), 19 check digit, 169, 171 checksum, 171 credit cards, 173 UPC, 173

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332 INDEX

Chinese remainder theorem, 311 choice function, 237 Chomp game, 181 classe, 88 close method of file, 161 Cn, 279 coefficients of a polynomial, 205 combinations, 131, 140 comments, 15, 30 common divisor, 296 compiler, 25 complement of a set, 102 complete graph, 263 complex numbers, 213 conditional branching instructions, 20 conditional jump instructions, 105 conditional operators, 108 congruence modulo d, 306 conjunction, 96 connected graph, 268, 279 constants, 40 converging sequence, 58 converging series, 64 corollary, 213, 302 CPU (Central Processing Unit), 19 instructions, 20 registers, 21 Craps game, 233 cryptology, 313 cycle (graph), 264 database of sequences, 58 De Morgans laws, 97, 103 debug program, 20 debugging, 20 decimal into binary conversion, 83 def statement, 32 default values for arguments, 52 degree of a vertex in a graph, 270 del operator for lists, 155 dictionary, 244, 254

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difference operator for sets, 252 Diffie-Hellman Key Exchange

algorithm, 314 digital electronics, 117 Diophantine equation, 297 directed graphs, 180, 274, 279 directories, 160 discard method of set, 253 discrete logarithm, 316 disjunction, 96 diverging sequence, 58 division operator, 53 divisor, 301 divisors of zero, 309 documentation string, 32, 33 domain of a function, 3 dot product of vectors, 248 edges of a graph, 259, 262 elements of a set, 2 elif keyword, 107 else clause, 106 empty set, 2, 251 endswith method of string, 152 equivalence relation, 267, 269, 306 class, 267 Euclids algorithm, 296 Euler circuit, 272, 278 Euler path, 260, 272 Euler, Leonhard, 259 even parity, 169, 170 exceptions, 40, 73 exponent (in floating-point numbers),

88 factor theorem for polynomials, 213 factorial function, 74 False and True constants, 107 Fermats Last Theorem, 295 Fermats Little Theorem, 310, 317

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INDEX 333

Fibonacci numbers, 173, 190, 193, 203, 301

file, 160 ASCII, 160 binary, 160 closing, 161 creating, 163 extension, 160 folder, 160 opening, 161 output, 163 readlines method, 162 text, 160 write method, 163 file object type, 162 find method of string, 149, 150 finite set, 2 finite strategy game, 169, 179 float type, 38 floating-point numbers, 38 IEEE standard, 88 flowchart, 12 for loop, 147 formal parameters, 51 formatted output, 40 Four-Color theorem, 287 Fraction class example, 88 fully triangulated graph, 288 function names, 15, 31 function, 1 arguments, 2, 46, 51 defined with a formula, 7 domain, 3 formal parameters, 51 natural domain, 7 range, 3 recursively defined, 189 raising an exception, 48 fundamental theorem of algebra, 213 fundamental theorem of arithmetic,

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301, 304

garbage collection, 150 gates, 117, 118 general term of a sequence, 57 geometric mean, 59 geometric sequence, 59 common difference, 58 sum of, 61 geometric series, 64 gigabyte, 19, 20 gigahertz, 20 global variables, 44 Goldbachs conjecture, 305 golden ratio, 93, 191 graph, 259 adjacency matrix, 280 complete, 263 connected, 268 cycle, 264 degree of a vertex, 270 directed, 274 isomorphism, 266 multigraphs, 262 optimal path, 275 planar, 262 properly colored, 284 simple, 262 subgraph, 269 weighted, 275 greatest common divisor, 296 Greco-Roman square, 80 Hamilton circuit, 273 Hamilton, Sir William Rowan, 273 harmonic series, 65 hash table, 244, 251, 254 hashing, 251 hex built-in function, 84 hex digits, 128 hex into binary conversion, 82 hex numbers, 124 in Python, 83 hexadecimal system, 21, 82

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334 INDEX

IDLE program, 27, 31 IEEE standard for floating-point

numbers, 88 if-else statement, 105 elif keyword, 107 else clause, 106 with pass, 106 imaginary number 1i = , 213 immutable objects, 41, 53,149, 154,

157 import statement, 45 in operator, 108 for strings, 148 for lists, 154 for tuples, 157 indentation, 14 indentation, 32 independent events, 233 index method of list, 155 indices, 148 negative, 148 infinite set, 2 infinity symbol, 64 input function, 48, 72 insert method of list, 154 int built-in function, 84, 123 int type, 38 interpreter, 25 intersection of sets, 101 intersection operator for sets, 252 invariant, 169, 175 irrational numbers, 92 ISBN-10 and ISBN-13 checkdigit,

174 isomorphic graphs, 266 isomorphism, 179 issubset method of set, 253 iterations, 69, 190 iterative statements, 69 Java, 20

Kempes chains, 290 Keno game, 230 keyword, 31 kilobyte, 20 Kn, 279 Koch Snowflake, 67 laws of logic, 97 duality, 97 len built in function, 148, 252 limit of the sequence, 58 linear algebra, 243 linear Diophantine equation, 298 linear polynomial, 206 list built-in function, 154 list comprehensions, 156 lists, 154 reversing, 155 sorting, 154 literal string, 17, 29 local variables, 43 logical operators, 69, 107, 108 long division for polynomials, 212 loop invariant, 176 Lucas, Edouard, 200 Mad Libs program, 167 magic square, 80, 250 mantissa, 88 map, 254 mapping, 1 Mastermind game, 159 math module, 45 mathematical induction, 62, 135, 200,

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288 matrix, 243, 245 max built in function, 154, 252 megabyte, 20 Mersenne prime, 113 methods, 48, 149

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INDEX 335

min built in function, 154, 252 mod operator, 305 Monte Carlo simulations, 238 motherboard, 19 multigraphs, 262, 279, 281 multiplication of polynomials, 210 multiplication rule, 131 for probabilities, 231 names in Python, 31 of variables, 39 NAND gate, 120 natural domain, 7 n-choose-k numbers, 140 negation, 97 nested loops, 70, 72 newline character, 160, 161 Nim sum, 183 Nim, 182 NOR gate, 120 NOT circuit, 116 NOT gate, 118 not in operator, 108 not logical operator, 69, 107 octal numbers, 124 in Python, 83 octal system, 81 odd parity, 170 OOP (object-oriented programming),

149 open built-in function, 161 Open Source license, 27 operating system, 160 optimal path (in a weighted graph),

275 OR circuit, 116 OR gate, 118 or logical operator, 69, 107 output formatting, 40

parity bit, 169, 170 parity, 170 parsing, 28 partial sum, 64 Pascal, Blaise, 131, 217 Pascals Triangle, 217 pathname, 161 perfect number, 75 permutations, 135 planar graphs, 262, 284 polynomials, 205 addition, 206 coefficients, 205 division with a remainder, 211 factor theorem, 213 linear, 206 long division, 212 multiplication, 210 quadratic, 206 remainder theorem, 213 roots (zeros), 213 subtraction, 208 pop CPU instruction, 195 positional number system, 78 postfix notation, 150 precedence of ope

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